What happens to the original text app threads on the phone once Anywhere is operational…or is the app just downloaded to the computer and then syncs with the phone’s text app? Do old texts transfer or do you start threads from scratch?

Once the Republic Anywhere is installed on your phone the most recent (about a month) is backed up to the sever and is accessible to the Window/Mac/Linux/(soon to be iOS)
for best results one should make the anywhere app the default SMS app on the phone but if you wish one can still use a different phone app (note you will not be able to see text sent from the computer if you are not using Republic anywhere phone app)

as for what happens to the default app it just sit there, as part of the ROM there no real savings by uninstalling it